If someone offer his/her help, is it possible to answer in polite form like that - "I can manage myself"?
You can say one of the following (in your most humble and polite voice ;) ):
- Thank you for coming forward to help me, but I am fine.
- Oh, thank you so much, but I am fine. (or) Oh! Thank you so much! But I am fine.
- I appreciate your help, but I am good.
- I appreciate your help, but I am fine.
- I appreciate your help, but I can manage myself.
In this context, myself means by myself (on my own, without assistance, etc.).
To manage oneself can mean to personally handle one's own particular circumstances and problems (insofar as they need to be "managed"). This emphasises the fact of the problems being "personal".
It's my opinion that adding by shifts the emphasis more to the fact of not needing assistance, but actually both forms are often used with this sense. It's also my opinion that both forms are very often spoken in a dismissive/hostile context - you actually want the other person to leave you alone to do whatever you need to on your own. Such a response is often followed by a sarcastic "thank you!" or "thank you very much!"
Leading on from that, I would be inclined to say that...
"I can manage on my own, thank you"
...carries less risk of the hearer feeling that his offer of help has been somewhat rudely rejected.